In ∆ABC AD is the median through A and E is the midpoint of AD . BE is produced to meet AC in F. Proved thatAF=1/3 AC.

Given: AD is the median of triangle. ABC. E is the mid point of AD.
BE produced meets AD at F
To prove : AF = 1/3 AC

Construction = Through D, draw DG parallel to BF
Proof : In triangle ADG
E is the midpoint of AD and EF parallel DG
According to Reverse of Midpoint theorem
F becomes midpoint of AG and AF = FG-- ( i )
In triangle BCF
D is the midpoint of BC and DG parallel to BF
According to Reverse of Midpoint theorem
G becomes midpoint of CF and FG = GC -- ( ii )
From equation ( i ) & ( ii ),
AF = FG= GC = AC
Now, AF + FG + GC = AC
AF + AF+ AF = AC ( since AF = FG= GC )
3 AF = AC
AF = 1/3 AC